That happens if a https site has links to a http site, or a https site with not valid certificate and whatnot. As annoying as it may sound, it's for your safety, you have to confirm manually each time.

Of course, I'm just kidding, that depends if one trusts the site or not. I am on a certain forum everyday which has a self signed certificate, I trust it 100%, others could not want to touch it with a telegraph pole =)

Others might read it, make the browser insecure and then later claim that Vivaldi is insecure because it allowed to steal their passwords, bring their computers to the knees etc. pp.
Sadly people are like that.

Why this warning notification isn't present in Firefox on the same sites? Is it some false warn in Vivaldi?

Probably it's because Mozilla (Firefox's maker) maintains its own security certificate store that is used by Firefox for https authentication, whereas chromium-based browsers like Vivaldi rely on Windows' certificate store for that. It sounds as if the Windows cert for the site perhaps has not been included or updated in some way like the Mozilla cert may have been. In that case, the Windows cert(s) need to be updated, or cert-updating turned on in the OS if it's been turned off.